Stochastic Reduced-Order Representations for Parametric and Model-Form Uncertainties, with Applications to Fracture and Molecular Dynamics Simulations

Abstract

Uncertainty quantification (UQ) has become a central pillar in computational science. Uncertainties can arise from, e.g., intrinsic fluctuations in materials and/or model-form variability associated with simplified or competing representations of the governing physics. This dissertation addresses two fundamental UQ challenges: (i) the construction of efficient surrogate models to propagate uncertainties through expensive forward models, and (ii) the representation of model-form uncertainties when multiple plausible models coexist.To address the first challenge, we develop a statistical surrogate model that combines nonlinear dimensionality reduction with reduced-order Hamiltonian Monte Carlo sampling, used for data augmentation. Focusing on phase-field simulations of brittle fracture in random mesostructures, the approach is compared with state-of-the-art methods, including the Fourier Neural Operators and PCA-Net. The framework is shown to achieve similar accuracy as baseline techniques, at a fraction of the cost induced by training data generation. The second challenge is addressed through the lens of reduced-order modeling. The approach relies on the expansion of the approximation space, using a stochastic reduced-order basis. The latter is defined through stochastic Riemannian convex combinations on a tangent space, which enables sampling in the convex hull defined by a set of nominal model classes. Various refinements of the formulation are proposed, including the use of an augmented snapshot matrix, the definition of the reduced-order dimension through reconstruction errors, and the enrichment of the representation through nonlinear terms. Applications to molecular dynamics simulations (with machine-learned potentials) on battery materials are presented to demonstrate the relevance of the probabilistic modeling framework.By developing a statistical surrogate model and stochastic frameworks enabling the representation of model-form uncertainties, this dissertation advances the state-of-the-art in uncertainty quantification for computational mechanics, thereby enhancing the credibility and predictive power of simulations for materials modeling and design.